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## Homework Statement

(Note: This isn't an assignment problem, more a curiosity about the derivation of an equation - hopefully it is still posted in the right forum..)

I have working through the derivation for the partial molar Gibbs free energy of mixing from the Flory-Huggins expression for the Gibbs free energy of mixing, however my math skills are (

**very**) rusty - especially when it comes to partial derivatives.

## Homework Equations

Gibbs Free Energy of Mixing (Flory-Huggins Theory):

ΔG

^{M}=R·T[n

_{1}·ln(ϕ

_{1})+n

_{2}·ln(ϕ

_{2})+n

_{1}·

_{2}]

partial molar Gibbs free energy of dilution:

Δμ

_{1}=R·T[ln(1-ϕ

_{2})+(1-1/r)·ϕ

_{2}+

_{2}

^{2}]

where

K

_{B}=Boltzmann Constant,

n

_{i}=number of moles of i,

ϕ

_{i}=volume fraction of i,

ϕ

_{1}=N

_{1}/N

_{0}=N

_{1}/(N

_{1}+r·N

_{2})

ϕ

_{2}=(r·N

_{2})/N

_{0}=(r·N

_{2})/(N

_{1}+r·N

_{2})

N

_{0}=N

_{1}+r·N

_{2}

where

N

_{0}is the number of lattice sites

N

_{1}is the number of solvent molecules

N

_{2}is the number of polymer molecules, each occupying "r" lattice sites (or "r" segments)

and R=K

_{B}·N

_{A}

where

N

_{A}= Avogadro's constant

## The Attempt at a Solution

ΔG

^{M}=R·T[n

_{1}·ln(ϕ

_{1})+n

_{2}·ln(ϕ

_{2})+n

_{1}·

_{2}]

applying R=K

_{B}·N

_{A}

ΔG

^{M}=K

_{B}·T[N

_{1}·ln(ϕ

_{1})+N

_{2}·ln(ϕ

_{2})+N

_{1}·

_{2}]

expressing ϕ

_{1}and ϕ

_{2}in terms of N

_{1}and N

_{2}gives

ΔG

^{M}=K

_{B}·T[N

_{1}·ln(N

_{1}/(N

_{1}+r·N

_{2}))+N

_{2}·ln((r·N

_{2})/(N

_{1}+r·N

_{2}))+N

_{1}·

_{2})/(N

_{1}+r·N

_{2})]

Next step would be to take the partial derivative of the equation with respect to N

_{1}, I've tried many times but I cannot get anything close to the equation for the partial molar Gibbs free energy of dilution..

I broke it up, letting

Ψ

_{1}=N

_{1}·ln(N

_{1}/(N

_{1}+r·N

_{2}))

Ψ

_{2}=N

_{2}·ln((r·N

_{2})/(N

_{1}+r·N

_{2}))

Ψ

_{3}=N

_{1}·

_{2})/(N

_{1}+r·N

_{2})

taking the partial derivative

dΨ

_{1}/dN

_{1}=ln(N

_{1}/(N

_{1}+rN

_{2})+[1-(N

_{1}+rN

_{1}N

_{2})/(N

_{1}+rN

_{2})]

dΨ

_{2}/dN

_{1}=-N

_{2}[(1+rN

_{2})/(N

_{1}+rN

_{2})]

dΨ

_{2}/dN

_{1}=

_{2})/(N

_{1}+rN

_{2})]+N

_{1}

_{2})/(N

_{1}+rN

_{2})

^{2}

which when I plug it all back in, gives me a big mess..

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